Theorem keephyp2v

Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth ). (Contributed by NM, 16-Apr-2005)

	Ref	Expression
Hypotheses	keephyp2v.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜓 ↔ 𝜒 ) )
	keephyp2v.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜒 ↔ 𝜃 ) )
	keephyp2v.3	⊢ ( 𝐶 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜏 ↔ 𝜂 ) )
	keephyp2v.4	⊢ ( 𝐷 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜂 ↔ 𝜃 ) )
	keephyp2v.5	⊢ 𝜓
	keephyp2v.6	⊢ 𝜏
Assertion	keephyp2v	⊢ 𝜃

Proof

Step	Hyp	Ref	Expression
1		keephyp2v.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜓 ↔ 𝜒 ) )
2		keephyp2v.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜒 ↔ 𝜃 ) )
3		keephyp2v.3	⊢ ( 𝐶 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜏 ↔ 𝜂 ) )
4		keephyp2v.4	⊢ ( 𝐷 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜂 ↔ 𝜃 ) )
5		keephyp2v.5	⊢ 𝜓
6		keephyp2v.6	⊢ 𝜏
7		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐶 ) = 𝐴 )
8	7	eqcomd	⊢ ( 𝜑 → 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) )
9	8 1	syl	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
10		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐵 , 𝐷 ) = 𝐵 )
11	10	eqcomd	⊢ ( 𝜑 → 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) )
12	11 2	syl	⊢ ( 𝜑 → ( 𝜒 ↔ 𝜃 ) )
13	9 12	bitrd	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
14	5 13	mpbii	⊢ ( 𝜑 → 𝜃 )
15		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐶 ) = 𝐶 )
16	15	eqcomd	⊢ ( ¬ 𝜑 → 𝐶 = if ( 𝜑 , 𝐴 , 𝐶 ) )
17	16 3	syl	⊢ ( ¬ 𝜑 → ( 𝜏 ↔ 𝜂 ) )
18		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐵 , 𝐷 ) = 𝐷 )
19	18	eqcomd	⊢ ( ¬ 𝜑 → 𝐷 = if ( 𝜑 , 𝐵 , 𝐷 ) )
20	19 4	syl	⊢ ( ¬ 𝜑 → ( 𝜂 ↔ 𝜃 ) )
21	17 20	bitrd	⊢ ( ¬ 𝜑 → ( 𝜏 ↔ 𝜃 ) )
22	6 21	mpbii	⊢ ( ¬ 𝜑 → 𝜃 )
23	14 22	pm2.61i	⊢ 𝜃